The total displacement field $\mu $ for every point of a flexible body is obtained from the
displacement of a local frame defining the rigid motion of the body and from an
additional local displacement field ${w}_{L}$ corresponding to the small vibrations of the body.

(${G}_{0}$, ${G}_{1}$, ${G}_{2}$, and ${G}_{3}$) defines the global frame (${e}_{1}$, ${e}_{2}$, and ${e}_{3}$).

(${L}_{0}$, ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$) defines an orthonormal local frame.

$P$ is the rotation matrix from (${G}_{0}$, ${G}_{1}$, ${G}_{2}$, and ${G}_{3}$) to (${L}_{0}$, ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$).

The total displacement, $u$, can thus be expressed as:

Where, ${u}_{{L}_{0}}$, ${u}_{{L}_{1}}$, ${u}_{{L}_{2}}$, and ${u}_{{L}_{3}}$ are displacements of points ${L}_{0}$, ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$, respectively,

$X$, $Y$, and $Z$ are coordinates in the local frame (${L}_{0}$, ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$)

${u}_{R}$ is the rigid body contribution to the total
displacement

Local displacement is given by a combination of local vibration modes
${\text{\Phi}}_{L}^{i}$:

$${w}_{L}={\text{\Phi}}_{L}\alpha $$

Where, $\alpha $ is the vector of local modal contributions.

Rigid body displacement ${u}_{R}$ can also be expressed as a combination of 12
modes:

The choice of the local frame (${L}_{0}$, ${L}_{1}$, ${L}_{2}$, and ${L}_{3}$) is fully arbitrary. These points do not need to be
input explicitly. Their locations define local coordinates and thus, the components
of the modes ${\text{\Phi}}_{R}^{i}$.

If the flexible body contains elements with rotational DOF, three additional modes
must be added to the ${\text{\Phi}}_{R}^{i}$ family, accounting for the inertia associated with these DOF. The
components of these additional modes on each node of the flexible body having
rotational DOF are: